I have a box with $m$ coins, each with a probability $p_i$ for $1\leq i \leq m$ of flipping heads. As an experiment, I flip each coin $n_i$ times, recording the results.
What is a statistical test I can perform against the hypothesis that the coins are fair on average? (i.e., that $(\sum p_i)/m = 0.5$, or that if you choose a coin uniformly at random and flip it the probability of heads is 0.5)
For a single coin, you can use the binomial test, but I'm not sure how to adapt it for this situation -- it seems like you should be able to do so using some weighting scheme. Or, is there an approximation that would be appropriate? In my application, $m$ is fairly small, and the $n_i$ and $p_i$ are likely quite similar among themselves.